# Matlab tutorial for systems and control theory

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MATLAB is a programming language and data visualization software package which is especially effective in signal processing and systems analysis. This document is a brief introduction to MATLAB that focuses on those features that are of particular importance in 6.003.1 It is assumed that the reader is familiar with Project Athena, has an Athena account, and has little or no experience with MATLAB.

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## Nội dung Text: Matlab tutorial for systems and control theory

MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Electrical Engineering
and Computer Science

Signals and Systems — 6.003
INTRODUCTION TO MATLAB — Fall 1999

Thomas F. Weiss

Last modiﬁcation September 9, 1999

1
Contents
1 Introduction 3

2 Getting Started 3

3 Getting Help from Within MATLAB 4

4 MATLAB Variables — Scalars, Vectors, and Matrices 4
4.1 Complex number operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4.2 Generating vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3 Accessing vector elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5 Matrix Operations 5
5.1 Arithmetic matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5.2 Relational operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5.3 Flow control operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.4 Math functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6 MATLAB Files 7
6.1 M-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.1.1 Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.2 Mat-Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.3 Postscript Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.4 Diary Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Plotting 10
7.1 Simple plotting commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.2 Customization of plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

8 Signals and Systems Commands 11
8.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
8.2 Laplace and Z Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
8.3 Frequency responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
8.4 Fourier transforms and ﬁltering . . . . . . . . . . . . . . . . . . . . . . . . . 13

9 Examples of Usage 13
9.1 Find pole-zero diagram, bode diagram, step response from system function . 13
9.1.1 Simple solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
9.1.2 Customized solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
9.2 Locus of roots of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.3 Response of an LTI system to an input . . . . . . . . . . . . . . . . . . . . . 18

10 Acknowledgement 18

2
1 Introduction
MATLAB is a programming language and data visualization software package which is es-
pecially eﬀective in signal processing and systems analysis. This document is a brief in-
troduction to MATLAB that focuses on those features that are of particular importance in
6.003.1 It is assumed that the reader is familiar with Project Athena, has an Athena account,
and has little or no experience with MATLAB. Other MATLAB help is available through
Athena consulting which oﬀers a number of more tutorial handouts and short courses (ext.
3-4435), on-line consulting (type olc at the Athena prompt), and Athena on-line help (type
help at the Athena prompt). There are a number of books available that describe MAT-
LAB. For example, Engineering Problem Solving with Matlab, by D. M. Etter, published by
Prentice-Hall (1997) and Mastering MATLAB , by Hanselman and Littleﬁeld, published by
Prentice-Hall (1996). The paperback MATLAB Primer by K. Sigmon, published by CRC
Press (1994) is a handy summary of MATLAB instructions. Further information about
MATLAB can be found at the web page of the vendor (The MathWorks, Inc.) whose URL
is http://www.mathworks.com. Full documentation can be purchased by contacting The
MathWorks.

2 Getting Started
On Project Athena, MATLAB can be accessed directly from the Dashboard (menu at the
top of the screen after you login to Project Athena) by using the hierarchical menu and
navigating as follows:

Numerical/Math//Analysis and Plotting//MATLAB.

MATLAB will then open a command window which contains the MATLAB prompt ‘>>’.
MATLAB contains a number of useful commands that are similar to UNIX commands,
e.g., ‘ls’, ‘pwd’, and ‘cd’. These are handy for listing MATLAB’s working directory, checking
the path to the working directory, and changing the working directory. MATLAB checks
for MATLAB ﬁles in certain directories which are controlled by the command ‘path’. The
command ‘path’ lists the directories in MATLAB’s search path. A new directory can be
appended or prepended to MATLAB’s search path with the command path(path,p) or
path(p,path) where p is some new directory, for example, containing functions written by
the user.
There is specially designed software available which can also be accessed from the Project
Athena Dashboard by navigating as follows:

Courseware//Electrical Engineering and Computer Science//
6.003 Signals and Systems//MATLAB.

These commands display a graphical user interface for exploring several important topics in
6.003. The same software is used in lecture demonstrations.
1
Revisions of this document will be posted on the 6.003 homepage on the web.

3
3 Getting Help from Within MATLAB
If you know the name of a function which you would like to learn how to use, use the ‘help’
command:

>> help functionname

This command displays a description of the function and generally also includes a list of
related functions. If you cannot remember the name of the function, use the ‘lookfor’
command and the name of some keyword associated with the function:

>> lookfor keyword

This command will display a list of functions that include the keyword in their descriptions.
Other help commands that you may ﬁnd useful are ‘info’, ‘what’, and ‘which’. Descrip-
tions of these commands can be found by using the help command. MATLAB also contains
a variety of demos that can be with the ‘demo’ command.

4 MATLAB Variables — Scalars, Vectors, and Matri-
ces
MATLAB stores variables in the form of matrices which are M × N , where M is the number
of rows and N the number of columns. A 1 × 1 matrix is a scalar; a 1 × N matrix is a row
vector, and M ×1 matrix is a column vector. All elements of a matrix can be real or complex

numbers; −1 can be written as either ‘i’ or ‘j’ provided they are not redeﬁned by the user.
A matrix is written with a square bracket ‘[]’ with spaces separating adjacent columns and
semicolons separating adjacent rows. For example, consider the following assignments of the
variable x

Real scalar >> x = 5
Complex scalar >> x = 5+10j (or >> x = 5+10i)
Row vector >> x = [1 2 3] (or x = [1, 2, 3])
Column vector >> x = [1; 2; 3]
3 × 3 matrix >> x = [1 2 3; 4 5 6; 7 8 9]

There are a few notes of caution. Complex elements of a matrix should not be typed with
spaces, i.e., ‘-1+2j’ is ﬁne as a matrix element, ‘-1 + 2j’ is not. Also, ‘-1+2j’ is interpreted
correctly whereas ‘-1+j2’ is not (MATLAB interprets the ‘j2’ as the name of a variable.
You can always write ‘-1+j*2’.

4.1 Complex number operations
Some of the important operations on complex numbers are illustrated below

4
Complex scalar >> x = 3+4j
Real part of x >> real(x) =⇒ 3
Imaginary part of x >> imag(x) =⇒ 4
Magnitude of x >> abs(x) =⇒ 5
Angle of x >> angle(x) =⇒ 0.9273
Complex conjugate of x >> conj(x) =⇒ 3 - 4i

4.2 Generating vectors
Vectors can be generated using the ‘:’ command. For example, to generate a vector x that
takes on the values 0 to 10 in increments of 0.5, type the following which generates a 1 × 21
matrix

>> x = [0:0.5:10];

Other ways to generate vectors include the commands: ‘linspace’ which generates a vector
by specifying the ﬁrst and last number and the number of equally spaced entries between
the ﬁrst and last number, and ‘logspace’ which is the same except that entries are spaced
logarithmically between the ﬁrst and last entry.

4.3 Accessing vector elements
Elements of a matrix are accessed by specifying the row and column. For example, in the
matrix speciﬁed by A = [1 2 3; 4 5 6; 7 8 9], the element in the ﬁrst row and third
column can be accessed by writing

>> x = A(1,3) which yields 3

The entire second row can be accessed with

>> y = A(2,:) which yields [4 5 6]

where the ‘:’ here means “take all the entries in the column”. A submatrix of A consisting
of rows 1 and 2 and all three columns is speciﬁed by

>> z = A(1:2,1:3) which yields [1 2 3; 4 5 6]

5 Matrix Operations
MATLAB contains a number of arithmetic, relational, and logical operations on matrices.

5
5.1 Arithmetic matrix operations
The basic arithmetic operations on matrices (and of course scalars which are special cases
of matrices) are:

- subtraction
* multiplication
/ right division
\ left division
^ exponentiation (power)
’ conjugate transpose

An error message occurs if the sizes of matrices are incompatible for the operation. Division
is deﬁned as follows: The solution to A ∗ x = b is x = A\b and the solution to x ∗ A = b is
x = b/A provided A is invertible and all the matrices are compatible.
Addition and subtraction involve element-by-element arithmetic operations; matrix mul-
tiplication and division do not. However, MATLAB provides for element-by-element opera-
tions as well by prepending a ‘.’ before the operator as follows:

.* multiplication
./ right division
.\ left division
.^ exponentiation (power)
.’ transpose (unconjugated)
The diﬀerence between matrix multiplication and element-by-element multiplication is
seen in the following example

>>A = [1 2; 3 4]
A =
1 2
3 4
>>B=A*A
B =
7 10
15 22
>>C=A.*A
C =
1 4
9 16

5.2 Relational operations
The following relational operations are deﬁned:

6
< less than
greater than
>= greater than or equal to
== equal to
~= not equal to

These are element-be-element operations which return a matrix of ones (1 = true) and zeros
(0 = false). Be careful of the distinction between ‘=’ and ‘==’.

5.3 Flow control operations
MATLAB contains the usual set of ﬂow control structures, e.g., for, while, and if, plus
the logical operators, e.g., & (and), | (or), and ~ (not).

5.4 Math functions
MATLAB comes with a large number of built-in functions that operate on matrices on an
element-by element basis. These include:

sin sine
cos cosine
tan tangent
asin inverse sine
acos inverse cosine
atan inverse tangent
exp exponential
log natural logarithm
log10 common logarithm
sqrt square root
abs absolute value
sign signum

6 MATLAB Files
There are several types of MATLAB ﬁles including ﬁles that contain scripts of MATLAB
commands, ﬁles that deﬁne user-created MATLAB functions that act just like built-in MAT-
LAB functions, ﬁles that include numerical results or plots.

7
6.1 M-Files
MATLAB is an interpretive language, i.e., commands typed at the MATLAB prompt are
interpreted within the scope of the current MATLAB session. However, it is tedious to type
in long sequences of commands each time MATLAB is used to perform a task. There are two
means of extending MATLAB’s power — scripts and functions. Both make use of m-ﬁles
(named because they have a .m extension and they are therefore also called dot-m ﬁles)
created with a text editor like emacs. The advantage of m-ﬁles is that commands are saved
and can be easily modiﬁed without retyping the entire list of commands.

6.1.1 Scripts
MATLAB script ﬁles are sequences of commands typed with an editor and saved in an m-ﬁle.
To create an m-ﬁle using emacs, you can type from Athena prompt

athena% emacs filename.m &

or from within MATLAB

>> ! emacs filename.m &

Note that ‘!’ allows execution of UNIX commands directly. In the emacs editor, type
MATLAB commands in the order of execution. The instructions are executed by typing the
ﬁle name in the command window at the MATLAB prompt, i.e., the m-ﬁle ﬁlename.m is
executed by typing

>> filename

Execution of the m-ﬁle is equivalent to typing the entire list of commands in the command
window at the MATLAB prompt. All the variables used in the m-ﬁle are placed in MAT-
LAB’s workspace. The workspace, which is empty when MATLAB is initiated, contains all
the variables deﬁned in the MATLAB session.

6.1.2 Functions
A second type of m-ﬁle is a function ﬁle which is generated with an editor exactly as the
script ﬁle but it has the following general form:

function [output 1, output 2] = functionname(input1, input2)
%
%[output 1, output 2] = functionname(input1, input2) Functionname
%
% Some comments that explain what the function does go here.
%

8
MATLAB command 1;
MATLAB command 2;
MATLAB command 3;

The name of the m-ﬁle for this function is functionname.m and it is called from the MATLAB
command line or from another m-ﬁle by the following command

>> [output1, output2] = functionname(input1, input2)

Note that any text after the ‘%’ is ignored by MATLAB and can be used for comments.
Output typing is suppressed by terminating a line with ‘;’, a line can be extended by typing
‘...’ at the end of the line and continuing the instructions to the next line.

6.2 Mat-Files
Mat-ﬁles (named because they have a .mat extension and they are therefore also called dot-
mat ﬁles) are compressed binary ﬁles used to store numerical results. These ﬁles can be
used to save results that have been generated by a sequence of MATLAB instructions. For
example, to save the values of the two variables, variable1 and variable2 in the ﬁle named
ﬁlename.mat, type

>> save filename.mat variable1 variable2

Saving all the current variables in that ﬁle is achieved by typing

>> save filename.mat

A mat-ﬁle can be loaded into MATLAB at some later time by typing

6.3 Postscript Files
Plots generated in MATLAB can be saved to a postscript ﬁle so that they can be printed at
a later time (for example, by the standard UNIX ‘lpr’ command). For example, to save the
current plot type

>> print -dps filename.ps

The plot can also be printed directly from within MATLAB by typing

>> print -Pprintername

9
0.4

0.3

0.2

0.1
Amplitude

0
Figure 1: Example of the plotting of the
-0.1 function x(t) = te−t cos(2π4t).
-0.2

-0.3

-0.4
0 1 2 3 4 5 6 7 8
Time (s)

Type ‘help print’ to see additional options.

6.4 Diary Files
A written record of a MATLAB session can be kept with the diary command and saved
in a diary ﬁle. To start recording a diary ﬁle during a MATLAB session and to save it in
ﬁlename, type

>> diary filename

To end the recording of information and to close the ﬁle type

>> diary off

7 Plotting
MATLAB contains numerous commands for creating two- and three-dimensional plots. The
most basic of these commands is ‘plot’ which can have multiple optional arguments. A
simple example of this command is to plot a function of time.

t = linspace(0, 8, 401); %Define a vector of times from ...
0 to 8 s with 401 points
x = t.*exp(-t).*cos(2*pi*4*t); %Define a vector of x values
plot(t,x); %Plot x vs t
xlabel(’Time (s)’); %Label time axis
ylabel(’Amplitude’); %Label amplitude axis

This script yields the plot shown in Figure 1.

10
7.1 Simple plotting commands
The simple 2D plotting commands include

plot Plot in linear coordinates as a continuous function
stem Plot in linear coordinates as discrete samples
loglog Logarithmic x and y axes
semilogx Linear y and logarithmic x axes
semilogy Linear x and logarithmic y axes
bar Bar graph
errorbar Error bar graph
hist Histogram
polar Polar coordinates

7.2 Customization of plots
There are many commands used to customize plots by annotations, titles, axes labels, etc.
A few of the most frequently used commands are

xlabel Labels x-axis
ylabel Labels y-axis
title Puts a title on the plot
grid Adds a grid to the plot
gtext Allows positioning of text with the mouse
text Allows placing text at speciﬁed coordinates of the plot
axis Allows changing the x and y axes
figure Create a ﬁgure for plotting
figure(n) Make ﬁgure number n the current ﬁgure
hold on Allows multiple plots to be superimposed on the same axes
hold off Release hold on current plot
close(n) Close ﬁgure number n
subplot(a,b,c) Create an a × b matrix of plots with c the current ﬁgure
orient Specify orientation of a ﬁgure

8 Signals and Systems Commands
The following commands are organized by topics in signals and systems. Each of these
commands has a number of options that extend its usefulness.

8.1 Polynomials
Polynomials arise frequently in systems theory. MATLAB represents polynomials as row
vectors of polynomial coeﬃcients. For example, the polynomial s2 + 4s − 5 is represented in

11
MATLAB by the polynomial >> p = [1 4 -5]. The following is a list of the more impor-
tant commands for manipulating polynomials.

roots(p) Express the roots of polynomial p as a column vector
polyval(p,x) Evaluate the polynomial p at the values contained in the vector x
conv(p1,p2) Computer the product of the polynomials p1 and p2
deconv(p1,p2) Compute the quotient of p1 divided by p2
poly2str(p,’s’) Display the polynomial as an equation in s
poly(r) Compute the polynomial given a column vector of roots r

8.2 Laplace and Z Transforms
Laplace transforms are an important tool for analysis of continuous time dynamic systems,
and Z transforms are an important tool for analysis of discrete time dynamic systems. Im-
portant commands to manipulate these transforms are included in the following list.

residue(n,d) Compute the partial fraction expansion of the ratio of polynomials
n(s)/d(s)
lsim(SYS,u) Compute/plot the time response of SYS to the input vector u
step(SYS) Compute/plot the step response of SYS
impulse(SYS) Compute/plot the impulse response of SYS
pzmap(n,d) Compute/plot a pole-zero diagram of SYS
residuez(n,d) Compute the partial fraction expansion of the ratio of polynomials
n(z)/d(z) written as functions of z −1
rlocus(SYS) Compute/plot the root locus for a system whose open loop system is
speciﬁed by SYS
dlsim(n,d,u) Compute the time response to the input vector u of the system with
system function n(z)/d(z)
dstep(n,d) Compute the step response of the system with system function
n(z)/d(z)
dimpulse(n,d) Compute the impulse response of the system with system function
n(z)/d(z)
zplane(z,p) Plot a pole-zero diagram from vectors of poles and zeros, p and z

A number of these commands work with several speciﬁcations of the LTI system. One
such speciﬁcation is in terms of the transfer function for which ‘SYS’ is replaced with
‘TF(num,den)’ where ‘num’ and ‘den’ are the vectors of coeﬃcients of the numerator and
denominator polynomials of the system function.

12
8.3 Frequency responses
There are several commands helpful for calculating and plotting frequency response given
the system function for continuous or discrete time systems as ratios of polynomials.

bode(n,d) Plot the Bode diagram for a CT system whose system function is a the
ratio of polynomials n(s)/d(s)
freqs(n,d) Compute the frequency response for a CT system with system function
n(s)/d(s)
freqz(n,d) Compute the frequency response for a DT system with system function
n(z)/d(z)

8.4 Fourier transforms and ﬁltering
There is a rich collection of commands related to ﬁltering. A few basic commands are listed
here.

fft(x) Compute the discrete Fourier transform of the vector x
ifft(x) Compute the inverse discrete Fourier transform of the vector x
fftshift Shifts the ﬀt output from the discrete range frequency range (0, 2π) to
filter(n,d,x) Filters the vector x with a ﬁlter whose system function is n(z)/d(z),
includes some output delay
filtfilt(n,d,x) Same as filter except without the output delay

In addition there are a number of ﬁlter design functions including firls, firl1, firl2,
invfreqs, invfreqz, remez, and butter. There are also a number of windowing functions
including boxcar, hanning, hamming, bartlett, blackman, kaiser, and chebwin.

9 Examples of Usage
9.1 Find pole-zero diagram, bode diagram, step response from
system function
Given a system function
s
H(s) = ,
s2
+ 2s + 101
MATLAB allows us to obtain plots of the pole-zero diagram, the Bode diagram, and the
step response.

13
10 0 0.1

8 -20
0.08

Gain dB
6 -40
0.06

4 -60
0.04
2 -80 -1 0 1 2

Amplitude
Imag Axis

10 10 10 10 0.02
0
0
-2 90

-0.02
-4

Phase deg
0 -0.04
-6

-8 -0.06

-90
-10 -1 0 1 2 -0.08
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 10 10 10 10 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Real Axis Frequency (rad/sec) Time (secs)

Figure 2: The plots were generated by the script and each plot was saved as an encapsulated
postscript ﬁle. The three ﬁles were scaled by 0.3 and included in the document.

9.1.1 Simple solution
The simplest way to obtain the requisite plots is to let MATLAB choose all the scales, labels,
and annotations with the following script.
num = [1 0]; %Define numerator polynomial
den = [1 2 101]; %Define denominator polynomial

%% Pole-zero diagram
figure(1) %Create figure 1
pzmap(num,den); %Plot pole-zero diagram in figure 1

%% Bode diagram
figure(2); %Create figure 2
bode(num,den); %Plot the Bode diagram in figure 2

%% Step response
figure(3); %Create figure 3
step(num,den); %Plot the step response in figure 3
This script leads to the individual plots shown in Figure 2.

9.1.2 Customized solution
It may be desirable to display the results with more control over the appearance of the plot.
The following script plots the same results but shows how to add labels, add titles, deﬁne
axes scales, etc.

%% Define variables
t = linspace(0,5,201); %Define a time vector with 201 ...
equally-spaced points from 0 to 5 s.
w = logspace(-1,3,201); %Define a radian frequency vector ...
with 201 logarithmically-spaced ...

14
points from 10^-1 to 10^3 rad/s
num = [1 0]; %Define numerator polynomial
den = [1 2 101]; %Define denominator polynomial
[poles,zeros] = pzmap(num,den); %Define poles to be a vector of the ...
poles and zeros to be a vector of ...
zeros of the system function
[mag,angle] = bode(num,den,w); %Define mag and angle to be the ...
magnitude and angle of the ...
frequency response at w
[y,x] = step(num,den,t); %Define y to be the step response ...
of the system function at t

%% Pole-zero diagram
figure(1) %Create figure 1
subplot(2,2,1) %Define figure 1 to be a 2 X 2 matrix ...
of plots and the next plot is at ...
position (1,1)
plot(real(poles),imag(poles),’x’,real(zeros),imag(zeros),’o’); %Plot ...
pole-zero diagram with x for poles ...
and o for zeros
title(’Pole-Zero Diagram’); %Add title to plot
xlabel(’Real’); %Label x axis
ylabel(’Imaginary’); %Label y axis
axis([-1.1 0.1 -12 12]); %Define axis for x and y

%% Bode diagram magnitude
subplot(2, 2, 2); %Next plot goes in position (1,2)
semilogx(w,20*log10(mag)); %Plot magnitude logarithmically in w ...
and in decibels in magnitude
title(’Magnitude of Bode Diagram’);
ylabel(’Magnitude (dB)’);
axis([0.1 1000 -60 0]);
grid;
subplot(2, 2, 4); %Next plot goes in position (2,2)
semilogx(w,angle); %Plot angle logarithmically in w and ...
linearly in angle
title(’Angle of Bode Diagram’);
ylabel(’Angle (deg)’);
axis([0.1 1000 -90 90]);
grid;

15
Pole-Zero Diagram Magnitude of Bode Diagram
0
10

Magnitude (dB)
5
-20

Imaginary 0

-40
-5

-10
-60 0 2
-1 -0.5 0 10 10
Step Response Angle of Bode Diagram
0.1

0.05 50

Angle (deg)
Amplitude

0 0

-0.05 -50

-0.1 0 2
0 2 4 6 10 10

Figure 3: This combined plot was generated by the script and saved as an encapsulated
postscript ﬁle which was scaled by 0.6 and included in the document.

%% Step Response %Next plot goes in position (2,1)
subplot(2, 2, 3);
plot(t,y); %Plot step response linearly in t and y
title(’Step Response’);
xlabel(’Time (s)’);
ylabel(’Amplitude’);
grid;

This script leads to the plot shown in Figure 3.

9.2 Locus of roots of a polynomial
In analyzing a system, it is often of interest to determine the locus of the roots of a polynomial
as some parameter is changed. A common example is to track the poles of a closed-loop
feedback system as the open loop gain is changed. For example, in the system shown in
Figure 4, the closed loop gain is given by Black’s formula as

Y (s) G(s)
H(s) = = .
X(s) 1 + KG(s)

If G(s) is a rational function in s then it can be expressed as

16
X(s) + Y (s)
+ G (s)

Figure 4: Feedback system with open loop gain
G(s).
K

20

15

10

5
Imag Axis

0
Figure 5: Root-locus plot of the poles of H(s)
-5 as K varies for G(s) = s/(s2 + 2s + 101).
-10

-15

-20
-20 -15 -10 -5 0 5 10 15 20
Real Axis

N (s)
G(s) =
D(s)
where N (s) and D(s) are polynomials. Solving for H(s) yields
N (s)
H(s) = .
D(s) + KN (s)
Finding the poles of H(s) as K changes implies ﬁnding the roots of the polynomial D(s) +
KN (s) as K changes. Because this is such a common computation, MATLAB provides a
function for computing the root locus conveniently called ‘rlocus’. The following script
plots the root locus for G(s)
s
G(s) = 2 .
s + 2s + 101
num = [1 0]; %Define numerator polynomial
den = [1 2 101]; %Define denominator polynomial
figure(1);
rlocus(num,den) %Plot root locus
The plot is shown in Figure 5. The root-locus plot can be customized in a manner similar
to the example given above, see ‘help rlocus’.
The command rlocus can be used in other contexts. For example, suppose we have a
series R, L, C circuit whose admittance is
1 s
Y (s) = 2 + Rs + 2
,
Ls
and we wish to obtain a plot of the locus of the poles (roots of the denominator polynomial)
as the resistance R varies. It is only necessary to parse the denominator polynomial into two
parts N (s) = s and D(s) = s2 + 2 and use ‘rlocus’ to obtain the plot.

17
1

0.8

0.6

0.4

0.2
Figure 6: The input time function is a cos-
Amplitude

0
inusoid that starts at t = 0. The out-
-0.2
put of an LTI system with system function
-0.4
H(s) = 5s/(s2 + 2s + 101) is shown in red.
-0.6

-0.8

-1
0 1 2 3 4 5 6 7 8 9 10
Time (s)

9.3 Response of an LTI system to an input
The MATLAB command lsim makes it easy to compute the response of an LTI system with
system function H(s) to an input x(t). Suppose
5s
H(s) = ,
s2 + 2s + 101
and we wish to ﬁnd the response to the input x(t) = cos(2πt) u(t). The following script
computes and plots the response.

figure(1);
num = [5 0]; %Define numerator polynomial
den = [1 2 101]; %Define denominator polynomial
t = linspace(0, 10, 401); %Define a time vector
u = cos(2*pi*t); %Compute the cosine input function
[y,x] = lsim(num,den,u,t); %Compute the response to the input u at times t
plot(t,y,’r’,t,u,’b’); %Plot the output in red and the input in blue
xlabel(’Time (s)’);
ylabel(’Amplitude’);

The plot is shown in Figure 6.

10 Acknowledgement
This document makes use of earlier documents prepared by Deron Jackson and by Alan
Gale.

18

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